Formulation and Preparation for Numerical
Evaluation of Linear Complementarity
Systems in Dynamics
CHRISTOPH GLOCKER and CHRISTIAN STUDER IMES - Center of Mechanics, ETH Zurich, CH-8092 Zurich, Switzerland; E-mail: christoph.glocker@imes.mavt.ethz.ch
(Received 4 February 2004; accepted in revised form 19 May, 2004)
Abstract. In this paper, we provide a full instruction on how to formulate and evaluate planar
fric-tional contact problems in the spirit of non-smooth dynamics. By stating the equations of motion as an equality of measures, frictional contact reactions are taken into account by Lagrangian multipliers. Contact kinematics is formulated in terms of gap functions, and normal and tangential relative ve-locities. Associated frictional contact laws are stated as inclusions, incorporating impact behavior in form of Newtonian kinematic impacts. Based on this inequality formulation, a linear complementarity problem in standard form is presented, combined with Moreau’s time stepping method for numerical integration. This approach has been applied to the woodpecker toy, of which a complete parameter list and numerical results are given in the paper.
Keywords: unilateral constraints, friction, impact, time stepping, linear complementary problem
1. Introduction
During the last years, increasing interest in the behavior of dynamic systems with discontinuities has been observed in nonlinear dynamics, comprising in particular friction and impact problems. As research in the past was primarily focussed on systems with a single non-smooth interaction element, current trends point towards more and more complex situations, involving combined frictional and unilateral impact behavior and multi-contact problems. The latter has been a topic of active research for years in the area of non-smooth mechanics, concerning e.g. the mod-eling and mathematical formulation of set-valued interaction laws via inclusions, existence and uniqueness results of the associated measure differential inclusions, as well as the development of numerical algorithms for solving both, inequality and evolution problems. This concise paper is intended to prepare some existing material in this field in a most comprehensive and condensed form, such that the interested reader should find a quick access to the methods used. Among many ap-proaches, we have chosen the concept of linear complementarity to embed a subclass of non-smooth dynamic systems in mechanics, i.e. multibody systems with planar frictional impact contacts. We present a theoretical framework that is sufficiently wide to treat fully finite-dimensional dynamic systems with planar unilateral
Coulomb-contact constraints. A second goal is to show a general strategy on how to set up linear complementarity problems in a most efficient way when set-valued relay functions are involved. This step is non-trivial. In this paper, an LCP in stan-dard form is formulated with closest possible connection to optimization theory and with a minimum of equations and operations for matrix inversion. An example that fits in with this class of problems is the woodpecker toy, which might serve as a benchmark problem for low-dimensional non-smooth systems. The paper is self-contained in the sense that everything needed to evaluate the woodpecker example, from the description of the model and a complete data list up to the numerical results obtained by the presented time stepping method, is included.
The paper is organized as follows: In Section 2 we introduce the two most basic set-valued maps defined onR, the unilateral primitive Upr(x) and the filled-in relay function Sgn (x). It is shown that the relay function can be decomposed into two unilateral primitives, which is later quintessential when the linear complementar-ity problem of the dynamic system with planar frictional contacts is formulated. The theoretical framework for the formulation of a frictional contact problem in finite freedom dynamics is briefly presented in Section 3. We follow exactly the work in [12], but narrow it already down to planar unilateral Coulomb contacts with Newtonian impacts to have later access to linear complementarity. We give a complete formulation of the dynamic process based on a measure differential inclusion and try to clearly arrange the full set of unilateral contact-impact laws. The woodpecker in Section 4 serves as a typical example for a low-dimensional dynamic system with frictional unilateral contacts. The full model is presented, together with a complete list of parameters that have been used for the numerical simulation. As results, we show the time history of the angular displacements and velocities of the woodpecker body and sleeve, and the associated limit cycles in the phase space. Section 5 reviews the midpoint rule that was introduced in [12] as the first time stepping algorithm for non-smooth dynamic systems. This discretization scheme has been used for the numerical integration of the woodpecker’s equation of motion to obtain the results shown in Section 4. In Section 6 we finally show how the linear complementarity problem has to be formulated for planar Coulomb contacts by using the discretized equations of Section 5. This procedure has been taken from [8] and has been adapted to account for impacts.
2. Basic Set-Valued Elements
A linear complementarity problem (LCP) is a problem of the following form [3, 15]: For given A ∈ Rn,n and b ∈ Rn, find x ∈ Rn and y ∈ Rn such that the linear equation y= Ax + b holds together with the complementarity conditions yi≥ 0,
xi≥ 0, yixi= 0 for i = 1, . . . , n. The latter conditions are often written in the form y 0, x 0, yTx= 0 or, equivalently, as 0 y ⊥ x 0.
It is convenient to introduce a maximal monotone set-valued map Upr defined on R+, which we call the unilateral primitive, and which is the most important
Figure 1. The maps x→ Upr(x) and x → Sgn (x).
multifunction related to complementarity, Upr(x) :=
{0} if x > 0
(−∞, 0] if x = 0 . (1)
The graph of this map is depicted in the left part of Figure 1. Apparently, we are now able to express each complementarity condition of the LCP by one inclusion, since
−y ∈ Upr(x) ⇔ y ≥ 0, x ≥ 0, xy = 0. (2)
Unilateral primitives are used in mechanics on displacement level and on velocity level to model unilateral geometric and kinematic constraints, such as free plays with stops, sprag clutches and the like. The associated set-valued force laws are conveniently stated as inclusions in the form (2).
A second maximal monotone set-valued map, frequently met in complementar-ity systems, is the filled-in relay function Sgn (x), defined by
Sgn (x) :=
{+1} if x > 0
[−1, +1] if x = 0
{−1} if x < 0
, (3)
see Figure 1 for the graph. Note the difference at x= 0 to the classical sgn-function, which is defined as sgn (x= 0) = 0. In mechanics, relay functions on velocity level are used to model any kind of dry friction. On displacement level, they describe the behavior of pre-stressed springs. The relay function (3) can be represented by two unilateral primitives as indicated in Figure 2, which yields in terms of inclusions
−y ∈ Sgn (x) ⇔ ∃ xR, xL such that −y ∈ +Upr(xR)+ 1 −y ∈ −Upr(xL)− 1 x = xR− xL . (4)
Figure 2. Decomposition of Sgn into Upr’s.
More details on this decomposition may be found in [8] together with various appli-cations of even more complex set-valued interaction laws and their representations via unilateral primitives and inclusions. By using Equation (2), we may finally express Equation (4) in terms of complementarities,
−y ∈ Sgn (x) ⇔ ∃ xR, xL s.t. 1+ y ≥ 0, xR ≥ 0, (1 + y) xR = 0 1− y ≥ 0, xL ≥ 0, (1 − y) xL = 0 x = xR− xL . (5)
This representation has to be used when a problem involving Sgn -multifunctions is formulated as an LCP in standard form.
3. Mechanical Systems with Planar Coulomb Friction
The most suitable way to express the Newton–Euler equations for non-smooth dynamics is in terms of an equality of measures as introduced in [12]. In addition to non-smooth impact-free motion, this formulation covers even impulsive behavior and should be considered as the starting point of any such problem in dynamics.
The following notation is used: I := [tA, tE] denotes a compact time interval
on which the motion of the system is of interest. Time is denoted by t, and the Lebesgue measure onR by dt. We investigate the dynamics of a mechanical system on an f -dimensional configuration manifold with frictional boundaries. The set of local coordinates in use is denoted by q ∈ Rf, and the associated velocities by u ∈ Rf. As functions of time, the velocities u: [tA, tE] → Rf are assumed to
be of bounded variation with differential measure du, leading to displacements
q(t)= q(tA)+tt
Au(τ) dτ that are absolutely continuous on I with ˙q = u almost
everywhere. We further denote the right and left limit of u(t) at t by u+(t) and u−(t), respectively, which might be different from each other in the case of an impact. For a thorough treatment of bounded variation functions, absolute continuity and measures we refer in particular to the article [13], or to any advanced text book on integration theory, such as [21] or [5]. The measure equality for such a dynamic
system reads as
M du− h dt − dR = 0, (6)
where M(q, t) is the symmetric and positive definite mass matrix of the system,
h(q, u, t) the f -tuple of the gyroscopic accelerations (Christoffel symbols) together
with all classical finite-valued generalized forces, and dR the force measure of pos-sibly atomic impact impulsions, in our case containing the contact forces. The terms
M and h can be derived, for example, by taking q as a set of classical generalized
coordinates of the system and evaluating Lagrange’s equations of second kind or the associated virtual work expressions. The resulting classical second order equa-tion M(q, t) ¨q − h(q, ˙q, t) = 0 would then describe the same system as above, but without any contacts and any contact forces.
We assume a total of n frictional unilateral constraints in the system, which are represented by n inequalities
gN i(q, t) ≥ 0, i = 1, . . . , n. (7)
The quantities gN i are the gap functions of the frictional contacts. They are
for-mulated such that gN i> 0 indicates an open contact with an Euclidean distance
of the contact points given by the value of gN i, gN i= 0 corresponds to a closed
contact, and gN i< 0 indicates forbidden overlapping or interpenetration. A detailed
description on how to define these inequalities in a multibody system may be found in [20]. We further introduce the set of active contacts
H(t) =i| gN i(q(t), t) = 0, (8)
which singles out the contacts at which contact forces may occur. The force measure dR in Equation (6) is therefore at most composed of the normal and tangential contact forces of the individuals i ∈ H and may be written as
dR=
i∈H
wN idN i + wT idT i. (9)
In this expression, (wN i, wT i)(q, t) are the generalized normal and tangential
force directions, and (dN i, dT i) the associated scalar normal and tangential
contact impulse measures of contact i . Note that integration over a singleton t gives the scalar impulsive contact forces, {t}(dN i, dT i)= (N i(t), T i(t)).
In case of impact-free motion, one obtains from the same expression that (dN i, dT i)= ( ˙N i, ˙T i) dt with ˙N i the normal and ˙T i the tangential scalar
contact force.
The system’s dynamics is not yet completely determined by Equations (6–9), because we still have to specify force laws that relate the contact impulse measures
(dN i, dT i) to the system’s kinematic state (q, u). To do so, we first introduce
the normal and tangential relative velocities in the contacts,
γN i = wTN iu+ ˆwN i, γT i = wTT iu+ ˆwT i (10)
with (wN i, wT i)(q, t) as in Equation (9) and ( ˆwN i, ˆwT i)(q, t) = (0, 0) only for
rheonomic systems, see e.g. [20] on how to obtain these terms. We choose for the normal direction of each contact a unilateral version of Newton’s impact law with local restitution coefficientεN i ∈ [0, 1], and for the tangential direction a Coulomb
type frictional law with friction coefficientµithat is complemented by a tangential
restitution behaviorεT i ∈ [0, 1]. We define
ξN i := γN i+ + εN iγN i−, ξT i := γT i++ εT iγT i−, (11)
where (γN i±, γT i±) := (γN i, γT i)(u±), and pose the normal and tangential impact
laws as
−dN i ∈ Upr(ξN i), −dT i ∈ µidN iSgn (ξT i). (12)
With Equations (6–12) we have now obtained a complete description of the dy-namics of the system, including both, impacts and impact-free motion.
Note that the impact laws (12) are actually impact-contact laws, because they hold for both, impacts and impact-free motion: In the first case, (dN i, dT i)
in Equation (12) has just to be replaced by the corresponding impulsive forces (N i, T i) (t) when integration over {t} has been performed. For the second
case, assume a time interval without impacts, i.e. a time interval in which the velocities are continuous u+= u−= u and the forces are non-impulsive (dN i, dT i)= ( ˙N i, ˙T i) dt. Under these assumptions, Equations (11) becomes
ξN i = (1 + εN i)γN i, ξT i = (1 + εT i)γT i,
which causes Equation (12) to be
− ˙N idt ∈ Upr((1 + εN i)γN i), − ˙T idt ∈ µi˙N idt Sgn ((1+ εT i)γT i).
With non-negative values of the restitution coefficients,εN i ≥ 0 and εT i ≥ 0, and
after “crossing out” dt from both inclusions, one obtains
− ˙N i ∈ Upr(γN i), − ˙T i ∈ µi˙N iSgn (γT i),
which are the force laws for impact-free motion of unilaterally constrained contacts with Coulomb friction [8, 20].
Figure 3. Woodpecker toy and mechanical model.
4. The Woodpecker Toy
A woodpecker hammering down a pole is a typical low-dimensional system com-bining unilateral constraints, impacts, Coulomb friction and jamming. The wood-pecker toy (Figure 3) consists of a pole, a sleeve with a hole that is slightly larger than the diameter of the pole, a spring and the woodpecker. In operation, the wood-pecker moves down the pole performing some kind of pitching motion, which is controlled by the sleeve. This mechanism of self-excitation may roughly be ex-plained as follows: Gravitation acts as an energy source. This energy is transmitted to the woodpecker and results in a vertical downward motion of the entire system. The woodpecker itself oscillates up and down. This oscillation interacts via the spring with the sleeve. It gains its energy from the downwards motion by turning the sleeve and switching on and off a frictional contact jamming of the sleeve at the pole. This mechanism ends up in a stable limit cycle with an energetic balance of the kinetic energy, gained per cycle by the falling height, and the dissipated energy due to the frictional contacts.
A planar model of the woodpecker toy is also shown in Figure 3. It consists of three rigid bodies: The woodpecker (center of mass S, mass mS, moment of inertia
JS), the sleeve (center of mass M, mass mM, moment of inertia JM), and the pole
that is fixed to the environment. The woodpecker and the sleeve are connected by a revolute joint with angular stiffness cϕ and are both under the influence of gravity
g. We equipped the system with three degrees of freedom that are the angular
displacement ϕS of the woodpecker, and the angular and vertical displacements
ϕM and y of the sleeve. Lateral deviations of the sleeve are small and thus not
by ωS, and the angular and vertical velocities of the sleeve by ωM and v. The
generalized coordinates q and associated velocities u may thus be stated as
q= ϕyM ϕS , u = ωvM ωS
with ˙q= u almost everywhere. (13)
The special geometrical design of the toy led us to assume only small deviations in the displacements during operation. Thus we set up the dynamic Equations (6– 10) based on a linearized kinematics. The mass matrix M and the vector h in Equation (6), M= mS+ mM mSlM mSlG mSlM JM+ mSl2M mSlMlG mSlG mSlMlG JS+ mSlG2 , h= −(mS+ mM)g −cϕ(ϕM− ϕS)− mSlMg −cϕ(ϕS− ϕM)− mSlGg , (14)
follow in a straightforward manner from the kinetic and potential energy of the system by working out Lagrange’s equations of second kind.
Altogether we take into account three different frictional contacts: Contact 1 is between the beak of the woodpecker and the pole. This contact constraint is not necessary for the woodpecker to work, but as beak impacts have been observed in reality, we want them to be included in our model. The more important contacts are between the sleeve and the pole: The diameter of the hole in the sleeve is slightly larger than the diameter of the pole. Due to the resulting clearance, the lower or upper edge of the sleeve may come into contact with the pole. This is modeled by the unilateral constraints 2 and 3. In particular, the lower sleeve contact 2 is most essential for the jamming mechanism to be switched on and off. The three gap functions (7) follow now directly from Figure 5,
gN 1= (lM+ lG− lS− rO)− hSϕS,
gN 2= (rM− rO)+ hMϕM,
gN 3= (rM− rO)− hMϕM.
(15)
The relative velocities (10) in the normal directions γN i are directly obtained by
differentiating the gap functions (15) with respect to time. In the tangential direc-tions, theγT i follow easily from Figure 5. The associated w vectors and ˆw scalars
that are also needed to set up the force measure dR in Equation (9) are wN 1= 00 −hS , wT 1= l1M lG− lS , ˆwN 1= ˆwT 1= 0, wN 2= h0M 0 , wT 2= r1M 0 , wˆN 2= ˆwT 2= 0, wN 3= −h0M 0 , wT 3= r1M 0 , wˆN 3= ˆwT 3= 0. (16)
Numerical integration of the system has been performed by applying the time step-ping method described in the next section. The parameters used together with the initial conditions are summarized in Table I. The numerical results in Figures 4–6 obtained by this method have been confirmed by at least three independent calcu-lations based on different numerical schemes. Among them was an event-driven evaluation which is published in [11], and which is complemented by a deep dis-cussion of the bifurcation behavior of the system.
For the woodpecker example, the maximal dimension of the associated linear complementarity problems is 6. This dimension results from at most two contacts that can be closed at the same time, with three equations each. This is due to the special woodpecker kinematics which excludes a simultaneous contact of the upper and lower edge of the sleeve. An LCP of dimension 6 requires 26= 64 different cases to be checked for consistency. Not all of them are physically admissible because of one slack variable per contact that has to be introduced for technical reasons. However, 16 different physical states are still possible. They result from any combination of four contact states of the contacts in the LCP, i.e. sticking, sep-aration, and two different directions of sliding. If a structured inequality approach like an LCP formulation is avoided, and the system is classically treated by a com-binatorial trial and error method, these 16 states have to be evaluated “by hand”. This requires already a certain effort, but becomes impracticable for problems of higher dimensions with some hundreds or thousands of contacts.
5. Time Discretization
One of the main topics of present research in non-smooth dynamics is the develop-ment of reliable numerical integration algorithms for measure differential inclusions from both the practical and the theoretical point of view. Much effort is put today in the further development of the so-called time-stepping methods, of which the first has been proposed as the midpoint rule in [12]. Time-stepping methods are
Table I. Parameters and initial conditions.
Geometry Radius of pole rO= 0.0025 m
Inner radius of sleeve rM= 0.0031 m
1 2Height of sleeve hM= 0.0058 m Distance M–G lM= 0.010 m Distance G–S lG= 0.015 m Distance S–P hS= 0.02 m Length of beak P–1 lS= 0.0201 m
Inertias Mass, sleeve mM= 0.0003 kg
Mass, woodpecker mS= 0.0045 kg
Moment of inertia, sleeve JM= 5.0 · 10−9kg m2
Moment of inertia, woodpecker JS= 7.0 · 10−7kg m2
Force elements Angular stiffness, spring cϕ= 0.0056 Nm rad−1
Gravity g= 9.81 m s−2
Contact parameters Normal restitution εN 1= 0.5
εN 2,3= 0
Tangential restitution εT 1,2,3= 0
Friction coefficients µ1,2,3= 0.3
Initial conditions Sleeve, displacement y(0)= 0 m
Sleeve, angle ϕM(0)= −0.1036 rad
Woodpecker, angle ϕS(0)= −0.2788 rad
Sleeve, velocity v(0) = −0.3411 m s−1 Sleeve, angular velocity ωM(0)= 0 rad s−1
Woodpecker, angular velocity ωS(0)= −7.4583 rad s−1
difference schemes including fully the complementarity conditions and the impact laws, by allowing a simultaneous treatment of impulsive and non-impulsive forces together with all inequalities involved.
In this section we review the original midpoint rule [12] that has been used for the numerical evaluation of the woodpecker toy. Generally, to perform numerical integration of a system (6–12) with respect to time, one has to address the following problem: For given initial time tAand known initial displacements qA := q (tA)∈
Rf and velocities uA := u (tA) ∈ Rf, find approximations of the displacements qE := q (tE) ∈ Rf and velocities uE := u (tE) ∈ Rf at the end tE of a chosen
time interval [tA, tE]. To apply the midpoint rule, the following steps have to be performed, see [14] for many helpful additional comments:
1. Choose a time step t and compute the midpoint tM := tA + 1
2t and the endpoint tE := tA+ t of the time interval
2. Compute the midpoint displacements qM := qA+1 2t · u
3. Matrix calculations:
(a) Compute M (qM, tM)∈ Rf, f and h (qM, uA, tM)∈ Rf
(b) For i= 1, . . . , n set up the index set H := { i | gN i(qM, tM)≤ 0 } with k
elements i1, . . . , ik (0≤ k ≤ n)
(c) For every i ∈ H compute wN i(qM, tM) ∈ Rf and ˆw
N i(qM, tM) ∈ R, as
well as wT i(qM, tM)∈ Rf and ˆwT i(qM, tM)∈ R
4. Computation of uE: In this step, equations (6), (9), (10), (11) and inclusions (12)
have to be solved: Find uE such that for every i ∈ H (6, 9): M (uE − uA)− h t − i∈H (wN iN i + wT iT i)= 0 (10): γN iE = wTN iuE+ ˆwN i γT iE = wTT iuE + ˆwT i (10): γN iA = wTN iuA+ ˆwN i γT iA = wTT iuA+ ˆwT i (11): ξN i = γN iE + εN iγN iA ξT i = γT iE + εT iγT iA (12): −N i ∈ Upr(ξN i) −T i ∈ µiN iSgn (ξT i) (17) 5. Computation of qE := qM+ 12t · uE ∈ Rf
More advanced discretization schemes may be found in the literature, such as the powerful -method in [9], an algorithm based on displacements with proven convergence [18, 19], or several other well-developed codes described in [22, 23, 2]. The inequality problem (17) can be solved numerically by many different meth-ods. For example, a Gauß–Seidel iteration is used in [14] in which the contacts are solved cyclically independent of each other. Another iterative algorithm, introduced in [16], splits the Coulomb contact problems into one overall normal and one overall tangential subproblem which are then alternately solved by optimization techniques until convergence. The associated variational formulations and convex minimiza-tion problems on acceleraminimiza-tion level for multibody systems can be found in [7] and [8]. A powerful way to directly solve (17) via a non-smooth Newton method is the Augmented Lagrangian approach which was introduced into mechanics by [1], see e.g. [10] for a review and [4] for the kinematical and mechanical formulation of the contact laws. Another possibility is to reformulate (17) as a complementarity problem in standard form and to invoke an appropriate solver. Such formulations may be found, among many others, in [22] and [17] along with proofs on existence and uniqueness of solutions. Depending on the choice of the numerical discretiza-tion scheme, the resulting complementarity problem will be linear as in our case, or nonlinear. Solution methods for nonlinear complementarity problems based on NCP-functions and a semi-smooth Newton algorithm are presented, for example, in [6].
Figure 4. Time history of the woodpecker’s angular displacement and velocity.
Figure 6. Limit cycle in the phase space.
In this paper, we choose to formulate (17) as a linear complementarity prob-lem in standard form, which will be done in the next section, and to invoke finally an appropriate LCP solver, based on the algorithms found in [3] and [15].
6. Setting up the LCP
The purpose of this section is to derive an LCP formulation of the inequality problem (17). This step is nontrivial and tricky, and must not be underestimated. The main difficulty stems from the Sgn -functions that have to be decomposed into Upr’s to achieve the desired complementarity formulation.
Many different approaches can be found in literature on how to perform this decomposition. Slack variables are introduced in various ways to express the relay function by complementarity conditions. Most of them, however, have serious dis-advantages with respect to the dimension or the structure of the resulting LCP. Pos-sible relations to optimization theory may easily be disguised when slack variables are introduced in an ad hoc manner. Other problems are that additional solutions, that means solutions that are not contained in the original problem and thus
contra-dict the original problem, might be generated when too many slack variables are
introduced without the necessary care, or that unnecessary matrix inversions have to be performed. Another attempt that is frequently undertaken and that does not solve the problem is to slightly shift the set-valuedness of the sgn-function away from the origin to a neighboring point.
We are convinced that our approach is the only reasonable one when an LCP representation in local variables is desired for systems involving relay functions, and that this approach is even general enough to be applied without any modifications for adjoint problems in electrical networks or hydraulics. The steps that we perform
to derive the LCP follow exactly the procedure described in [8] to approximate even spatial friction by friction pyramides. To set up the LCP, rigorous matrix notation is required. We define WN :=wN i1, . . . , wN ik , WT :=wT i1, . . . , wT ik ∈ Rf,k ˆ wN :=wˆN i1, . . . , ˆwN ik T , wTˆ :=wˆT i1, . . . , ˆwT ik T ∈ Rk ΛN := N i1, . . . , N ik T , ΛT := T i1, . . . , T ik T ∈ Rk γE N := γE N i1, . . . , γ E N ik T , γE T := γE T i1, . . . , γ E T ik T ∈ Rk γA N := γA N i1, . . . , γ A N ik T , γA T := γA T i1, . . . , γ A T ik T ∈ Rk ξN := ξN i1, . . . , ξN ik T , ξT := ξT i1, . . . , ξT ik T ∈ Rk N := diag εN i1, . . . , εN ik , T := diag εT i1, . . . , εT ik ∈ Rk,k µ := diagµi1, . . . , µik ∈ Rk,k
and rewrite the first four lines in (17) as
M (uE− uA)− h t − WNΛ N− WT ΛT = 0, γEN = WTNu E+ ˆw N, γET = WTT u E + ˆwT, γA N = WTNu A+ ˆwN, γA T = WTT u A+ ˆwT, ξN = γEN+ NγNA, ξT = γET + Tγ A T. (18)
The third line in Equation (18) is used to compute (γA N, γ
A
T) from the known
velocities uAat the left endpoint of the time interval and is thus no longer needed. The unknowns (γE
N, γET) in the fourth line may be immediately eliminated with the
help of the second line. This results in the reduced set of equations
M (uE − uA)− h t − WNΛN − WT ΛT = 0, (19) ξN = WTNu E +ˆ wN + NγAN , (20) ξT = WTTu E +wTˆ + T γTA , (21)
which will be used in the sequel to set up the linear complementarity problem. We are now going to formulate the inclusions from the last line in (17) as complementarity conditions. For the unilateral primitives we obtain by (2)
−N i ∈ Upr(ξN i) (i ∈ H)
ΛN 0, ξN 0, ΛTNξN = 0.
The relay functions have to be decomposed according to (5), but with a modified step height [−µiN i, +µiN i] which gives
−T i ∈ µiN iSgn (ξT i) (i ∈ H) ∃ ξR, ξL ∈ Rk such that µΛN + ΛT ≥ 0, ξR ≥ 0, (µΛN+ ΛT)TξR = 0 µΛN − ΛT ≥ 0, ξL ≥ 0, (µΛN − ΛT)TξL = 0 ξT = ξR− ξL . (23)
We further introduceΛR, ΛL ∈ Rk to abbreviate the complementarity conditions
in Equation (23). They are defined, together with the last equation in (23), as
ΛR := µΛN + ΛT, (24)
ΛL := µΛN− ΛT, (25)
ξT = ξR− ξL. (26)
The whole set of complementarity conditions (22), (23) therefore reads as
0 ξξNR ΛL ⊥ ΛΛNR ξL 0. (27)
Note the special arrangement ofΛL andξL in Equation (27). They must be placed
in this manner, which has deep roots in optimization theory. Otherwise, one is not able to set up the LCP without additional matrix inversion processes.
By Equation (27) we have already found the final description of the comple-mentarity conditions of our problem, which also defines the vectors of variables that are allowed to be used. In addition, we still have Equations (19–21) and (24– 26) with the unknownsΛN,R,L,T,ξN,R,L,T and uE. The magnitudesξT, ΛT and uE have to be eliminated from this set of six equations, because they are not con-tained as variables in Equation (27). In a first step, we use Equation (24) and Equation (26) to removeΛT andξT from the remaining equations. This yields for
Equations (19), (20), (21), (25) M (uE − uA)− ht − (WN− WTµ)ΛN − WTΛR = 0, (28) ξN = WTNu E +wNˆ + N γAN , (29) ξR= WTTu E +wTˆ + T γTA + ξL, (30) ΛL = 2 µ ΛN− ΛR. (31)
In a second step, we solve Equation (28) for uE, which is always possible due to the regularity of the mass matrix M,
uE = M−1(WN − WTµ) ΛN + M−1WTΛR+ M−1ht + uA, (32)
and plug it in Equations (29) and (30). As a result, we obtain (29–31) in matrix notation, ξξNR ΛL = WTNM−1(WN− WT µ) WTNM−1WT 0 WTTM−1(WN − WT µ) WTTM−1WT E 2µ −E 0 ΛΛNR ξL + WTNM−1ht + (E + N)γNA WTTM−1ht + (E + T)γTA 0 , (33) where we have used the third line in Equation (18) to express (WT
Nu A, WT Tu A) in terms of (γAN, γ A
T). Equation (33) together with (27) is now the desired LCP y= Ax + b, 0 y ⊥ x 0 of dimension 3k.
Note the following properties of the LCP (33), (27): For friction that is indepen-dent of the normal loadN i, the termsµiN i have to be replaced by constants ai
and move into the vector b of the LCP. The resulting matrix A is then bisymmetric, and the LCP states the Kuhn–Tucker conditions of an associated quadratic program with inequality constraints, see [8]. Fort = 0, the LCP reduces to the pure im-pact equations and can be used, for example, for initialization of the velocities. For
γA
N= γAT= 0, the LCP describes impact-free motion on velocity level, containing
still the cases of persisting contact and stiction as well as transitions to sliding or separation, and can be transformed to the acceleration level by the methods shown in [8].
Acknowledgments
Parts of this research was conducted within the European project SICONOS IST-2001-37172 and supported by the Swiss Federal Office for Education and Science (BBW).
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